For all \(d\) , the support \({\cal {T}}_{(d)}\)  and the \(\sigma \) -field \(\sigma \left({\cal {T}}_{(d)}\right)\)  is reduced by the possibility of early stopping: \(\sigma \left({\cal {T}}_{(d)}\right) \subset {\cal {F}}_{(d)}\) . This creates new measurable spaces \(\left({\cal {T}}_{(d)}, \sigma \left({\cal {T}}_{(d)}\right)\right)\)  for interim tests at every possible stage \(1,\ldots ,K\) . Armitage's recursive sub-density formula [[1]}] is defined on this measurable space; see Equation (REF ) for \(K=2\) . Re-scaling yields a density function [see Equation (REF )] which defines a probability measure to complete the probability space \(\left({\cal {T}}_{(d)}, \sigma \left({\cal {T}}_{(d)}\right), P_{(d)}\right)\) , where the probability measure \(P_{(d)}\)  is determined by density (REF ): \(P_{(d)}(A) = \int _A f_{T_{(D)}}(t|D\ge d,\theta ) dt\) , \(A \in \sigma \left({\cal {T}}_{(d)}\right)\) .
